THERE are certain mathematical realities associated with New York City private schools: There are more students than seats at the top-tier schools, at least three sets of twins will be vying head to head for spots in any class, and already-expensive tuition can only go up.

Way up.

Over the past 10 years, the median price of first grade in the city has gone up by 48 percent, adjusted for inflation, compared with a 35 percent increase at private schools nationally — and just 24 percent at an Ivy League college — according to tuition data provided by 41 New York City K-12 private schools to the National Association of Independent Schools.

Indeed, this year’s tuition at Columbia Grammar and Preparatory ($38,340 for 12th grade) and Horace Mann ($37,275 for the upper school) is higher than Harvard’s ($36,305). Those 41 schools (out of 61 New York City private schools in the national association) provided enough data to enable a 10-year analysis. (Over all, inflation caused prices in general to rise 27 percent over the past decade.)

The median 12th-grade tuition for the current school year was $36,970, up from $21,100 in 2001-2, according to the national association’s survey. Nationally, that figure rose to $24,240 from $14,583 a decade ago.